We study spaces of continuous functions and sections with domain a paracompact Hausdorff k-space $X$ and range a nilpotent CW complex $Y$, with emphasis on localization at a set of primes. For $\mathop {\rm map}\nolimits _\phi (X,\,Y)$, the space of maps with prescribed restriction $\phi$ on a suitable subspace $A\subset X$, we construct a natural spectral sequence of groups that converges to $\pi _*(\mathop {\rm map}\nolimits _\phi (X,\,Y))$ and allows for detection of localization on the level of $E^2$. Our applications extend and unify the previously known results.

We extend the group-theoretic notion of conditional flatness for a localization functor to any pointed category, and investigate it in the context of homological categories and of semi-abelian categories. In the presence of functorial fiberwise localization, analogous results to those obtained in the category of groups hold, and we provide existence theorems for certain localization functors in specific semi-abelian categories. We prove that a Birkhoff subcategory of an ideal determined category yields a conditionally flat localization, and explain how conditional flatness corresponds to the property of admissibility of an adjunction from the point of view of categorical Galois theory. Under the assumption of fiberwise localization, we give a simple criterion to determine when a (normal epi)-reflection is a torsion-free reflection. This is shown to apply, in particular, to nullification functors in any semi-abelian variety of universal algebras. We also relate semi-left-exactness for a localization functor L with what is called right properness for the L-local model structure.

For a subring $R$ of the rational numbers, we study $R$-localization functors in the local homotopy theory of simplicial presheaves on a small site and then in ${\mathbb {A}}^1$-homotopy theory. To this end, we introduce and analyze two notions of nilpotence for spaces in ${\mathbb {A}}^1$-homotopy theory, paying attention to future applications for vector bundles. We show that $R$-localization behaves in a controlled fashion for the nilpotent spaces we consider. We show that the classifying space $BGL_n$ is ${\mathbb {A}}^1$-nilpotent when $n$ is odd, and analyze the (more complicated) situation where $n$ is even as well. We establish analogs of various classical results about rationalization in the context of ${\mathbb {A}}^1$-homotopy theory: if $-1$ is a sum of squares in the base field, ${\mathbb {A}}^n \,{\setminus}\, 0$ is rationally equivalent to a suitable motivic Eilenberg–Mac Lane space, and the special linear group decomposes as a product of motivic spheres.

Let $G$ be a finite group with cyclic Sylow $p$-subgroups, and let $k$ be a field of characteristic $p$. Then $H^{*}(BG;k)$ and $H_*(\Omega BG{{}^{{}^{\wedge }}_p};k)$ are $A_\infty$ algebras whose structure we determine up to quasi-isomorphism.

For the $p$-localized sphere $\mathbb {S}^{2m-1}_{(p)}$ with $p >3$ a prime, we prove that the homotopy nilpotency satisfies $\mbox {nil}\ \mathbb {S}^{2m-1}_{(p)}<\infty$, with respect to any homotopy associative $H$-structure on $\mathbb {S}^{2m-1}_{(p)}$. We also prove that $\mbox {nil}\ \mathbb {S}^{2m-1}_{(p)}= 1$ for all but a finite number of primes $p >3$. Then, for the loop space of the associated $\mathbb {S}^{2m-1}_{(p)}$-projective space $\mathbb {S}^{2m-1}_{(p)}P(n-1)$, with $m,n\ge 2$ and $m\mid p-1$, we derive that $\mbox {nil}\ \Omega (\mathbb {S}^{2m-1}_{(p)}P (n-1))\le 3$.

Let X be a 4-dimensional toric orbifold. If $H^{3}(X)$ has a non-trivial odd primary torsion, then we show that X is homotopy equivalent to the wedge of a Moore space and a CW-complex. As a corollary, given two 4-dimensional toric orbifolds having no 2-torsion in the cohomology, we prove that they have the same homotopy type if and only their integral cohomology rings are isomorphic.

A directed space is a topological space $X$ together with a subspace $\vec {P}(X)\subset X^I$ of directed paths on $X$. A symmetry of a directed space should therefore respect both the topology of the underlying space and the topology of the associated spaces $\vec {P}(X)_-^+$ of directed paths between a source ($-$) and a target ($+$)—up to homotopy. If it is, moreover, homotopic to the identity map—in a directed sense—such a symmetry will be called an inessential d-map, and the paper explores the algebra and topology of inessential d-maps. Comparing two d-spaces $X$ and $Y$ ‘up to symmetry’ yields the notion of a directed homotopy equivalence between them. Under appropriate conditions, all directed homotopy equivalences are shown to satisfy a 2-out-of-3 property. Our notion of directed homotopy equivalence does not agree completely with the one defined in Goubault (2017, arxiv:1709:05702v2) and Goubault, Farber and Sagnier (2020, J. Appl. Comput. Topol. 4, 11–27); the deviation is motivated by examples. Nevertheless, directed topological complexity, introduced in Goubault, Farber and Sagnier (2020) is shown to be invariant under our notion of directed homotopy equivalence. Finally, we show that directed homotopy equivalences result in isomorphisms on the pair component categories of directed spaces introduced in Goubault, Farber and Sagnier (2020).

For certain pairs of Lie groups (G, H) and primes p, Harris showed a relation of the p-localized homotopy groups of G and H. This is reinterpreted as a p-local homotopy equivalence G ≃ (p)H × G/H, and so there is a projection G(p) → H(p). We show how much this projection preserves the higher homotopy associativity.

This paper concerns extension of maps using obstruction theory under a non-classical viewpoint. It is given a classification of homotopy classes of maps and as an application it is presented a simple proof of a theorem by Adachi about equivalence of vector bundles. Also it is proved that, under certain conditions, two embeddings are homotopic up to surgery if and only if the respective normal bundles are SO-equivalent.

For a prime $p$, let $\hat{F}_{p}$ be a finitely generated free pro-$p$-group of rank at least $2$. We show that the second discrete homology group $H_{2}(\hat{F}_{p},\mathbb{Z}/p)$ is an uncountable $\mathbb{Z}/p$-vector space. This answers a problem of A. K. Bousfield.

In this note, we have obtained a Whitehead-like tower of a module by considering a suitable set of morphisms and shown that the different stages of the tower are the Adams cocompletions of the module with respect to the suitably chosen set of morphisms.

Apart from simply connected spaces, a non-simply connected co-H-space is a typical example of a space X with a coaction of Bπ1 (X) along rX: X → Bπ1 (X), the classifying map of the universal covering. If such a space X is actually a co-H-space, then the fibrewise p-localization of rX (or the ‘almost’ p-localization of X) is a fibrewise co-H-space (or an ‘almost’ co-H-space, respectively) for every prime p. In this paper, we show that the converse statement is true, i.e. for a non-simply connected space X with a coaction of Bπ1 (X) along rX, X is a co-H-space if, for every prime p, the almost p-localization of X is an almost co-H-space.

The method for computing the $p$-localization of the group $\left[ X,\,\text{U}\left( n \right) \right]$, by Hamanaka in 2004, is revised. As an application, an explicit description of the self-homotopy group of $\text{Sp}\left( 3 \right)$ localized at $p\,\ge \,5$ is given and the homotopy nilpotency of $\text{Sp}\left( 3 \right)$ localized at $p\,\ge \,5$ is determined.

We study left and right Bousfield localisations of stable model categories which preserve stability. This follows the lead of the two key examples: localisations of spectra with respect to a homology theory and A-torsion modules over a ring R with A a perfect R-algebra. We exploit stability to see that the resulting model structures are technically far better behaved than the general case. We can give explicit sets of generating cofibrations, show that these localisations preserve properness and give a complete characterisation of when they preserve monoidal structures. We apply these results to obtain convenient assumptions under which a stable model category is spectral. We then use Morita theory to gain an insight into the nature of right localisation and its homotopy category. We finish with a correspondence between left and right localisation.

We construct a spectral sequence that computes the generalized homology E*(∏ Xα) of a product of spectra. The E2-term of this spectral sequence consists of the right derived functors of product in the category of E*E-comodules, and the spectral sequence always converges when E is the Johnson-Wilson theory E(n) and the Xα are Ln-local. We are able to prove some results about the E2-term of this spectral sequence; in particular, we show that the E(n)-homology of a product of E(n)-module spectra Xα is just the comodule product of the E(n)*Xα. This spectral sequence is relevant to the chromatic splitting conjecture.

Given a spectrum $X$, we construct a spectral sequence of $BP_{*}BP$-comodules that converges to $BP_{*}(L_{n}X)$, where $L_{n}X$ is the Bousfield localization of $X$ with respect to the Johnson–Wilson theory $E(n)_{*}$. The $E_{2}$-term of this spectral sequence consists of the derived functors of an algebraic version of $L_{n}$. We show how to calculate these derived functors, which are closely related to local cohomology of $BP_{*}$-modules with respect to the ideal $I_{n + 1}$.

Most extant localization theories for spaces, spectra and diagrams of such can be derived from a simple list of axioms which are verified in broad generality. Several new theories are introduced, including localizations for simplicial presheaves and presheaves of spectra at homology theories represented by presheaves of spectra, and a theory of localization along a geometric topos morphism. The $f$-localization concept has an analog for simplicial presheaves, and specializes to the ${{\mathbb{A}}^{1}}$-local theory of Morel-Voevodsky. This theory answers a question of Soulé concerning integral homology localizations for diagrams of spaces.

We show that, for a finite group G and a prime p, the following facts are equivalent: (i) the p-localization homomorphism l: G —> Gp induces p-localization on integral homology; (ii) the higher homotopy groups of the Bousfield-Kan Zp-completion of a K(G, 1) vanish; (iii) the group G is p-nilpotent.

In this paper p-local versions of the Rational Fibre and Cofibre Decomposition Theorems are given. In particular, if there exists an element in the nth Gottlieb group of a space F such that its image under the Hurewicz map has infinite order, then Sn for almost all primes p. A dual result is proved for cofibrations.